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Learning time-evolving objects such as multivariate time series and dynamicnetworks requires the development of novel knowledge representation mechanismsand neural network architectures, which allow for capturing implicittime-dependent information contained in the data. Such information is typicallynot directly observed but plays a key role in the learning task performance. Inturn, lack of time dimension in knowledge encoding mechanisms fortime-dependent data leads to frequent model updates, poor learning performance,and, as a result, subpar decision-making. Here we propose a new approach to atime-aware knowledge representation mechanism that notably focuses on implicittime-dependent topological information along multiple geometric dimensions. Inparticular, we propose a new approach, named \textit{Temporal MultiPersistence}(TMP), which produces multidimensional topological fingerprints of the data byusing the existing single parameter topological summaries. The main idea behindTMP is to merge the two newest directions in topological representationlearning, that is, multi-persistence which simultaneously describes data shapeevolution along multiple key parameters, and zigzag persistence to enable us toextract the most salient data shape information over time. We derivetheoretical guarantees of TMP vectorizations and show its utility, inapplication to forecasting on benchmark traffic flow, Ethereum blockchain, andelectrocardiogram datasets, demonstrating the competitive performance,especially, in scenarios of limited data records. In addition, our TMP methodimproves the computational efficiency of the state-of-the-art multipersistencesummaries up to 59.5 times.

Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistence